Removing Noise from a Seismic Measurement

ABSTRACT

A technique includes spatially filtering a signal that is derived from a seismic acquisition. The filtering is associated with a filter length, and the filtering includes varying the filter length with frequency. The filtering may be used in connection with adaptive noise attenuation, which is applied to decomposed subbands. Furthermore, the filtering may be applied during the reconstruction of the signal from the subbands.

BACKGROUND

The invention generally relates to removing noise from a seismicmeasurement.

Seismic exploration involves surveying subterranean geologicalformations for hydrocarbon deposits. A survey typically involvesdeploying seismic source(s) and seismic sensors at predeterminedlocations. The sources generate seismic waves, which propagate into thegeological formations creating pressure changes and vibrations alongtheir way. Changes in elastic properties of the geological formationscatter the seismic waves, changing their direction of propagation andother properties. Part of the energy emitted by the sources reaches theseismic sensors. Some seismic sensors are sensitive to pressure changes(hydrophones), others to particle motion (e.g., geophones and/oraccelerometers), and industrial surveys may deploy only one type ofsensors or both. In response to the detected seismic events, the sensorsgenerate electrical signals to produce seismic data. Analysis of theseismic data can then indicate the presence or absence of probablelocations of hydrocarbon deposits.

Some surveys are known as “marine” surveys because they are conducted inmarine environments. However, “marine” surveys may be conducted not onlyin saltwater environments, but also in fresh and brackish waters. In onetype of marine survey, called a “towed-array” survey, an array ofseismic sensor-containing streamers and sources is towed behind a surveyvessel.

SUMMARY

In an embodiment of the invention, a technique includes spatiallyfiltering a signal that is derived from a seismic acquisition. Thefiltering is associated with a filter length, and the filtering includesvarying the filter length with frequency.

In another embodiment of the invention, a technique includes decomposinga signal that is derived from a seismic acquisition into subbands insuccessive stages. The subbands are associated with at least differentfrequency ranges of the signal. The technique includes selectivelyapplying adaptive noise attenuation in between the successive stagessuch that the stages decompose noise-attenuated subbands; andreconstructing the signal from the subbands resulting from thedecomposition.

In another embodiment of the invention, a technique includes decomposinga signal that is derived from a seismic acquisition into subbands, whichare associated with at least different frequencies. The techniqueincludes reconstructing the signal from the subbands, includingspatially filtering the subband; and for each subband, regulating alength of the filtering based on the frequencies that are associatedwith the subband.

In another embodiment of the invention, a technique includes filtering asignal derived from a seismic acquisition and mitigating edge effects inconnection with the filtering. The mitigation includes usingautoregressive modeling to extrapolate the signal beyond an acquisitiongeometry to form an extrapolated signal and performing variable lengthspatial filtering on the extrapolated signal.

In yet another embodiment of the invention, a technique includesproviding an n dimensional finite impulse response (FIR) filter that isformed from a plurality of n−1 dimension FIR filters; and spatiallyfiltering a signal derived from a seismic acquisition in n dimensionsusing the plurality of n−1 dimension FIR filters to generate a filteredsignal.

Advantages and other features of the invention will become apparent fromthe following drawing, description and claims.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a schematic diagram of a marine-based seismic data acquisitionsystem according to an embodiment of the invention.

FIG. 2 is an illustration of a response of a low pass spatial filteraccording to an embodiment of the invention.

FIG. 3 is a plot representing a cutoff wavenumber of the filter of FIG.2 as a function of frequency according to an embodiment of theinvention.

FIG. 4 is a plot of a transition bandwidth of the filter of FIG. 2 as afunction of frequency according to an embodiment of the invention.

FIG. 5 is a plot of a filter length of the filter of FIG. 2 as afunction of frequency according to an embodiment of the invention.

FIGS. 6, 7 and 17 are flow diagrams depicting techniques to spatiallyfilter a signal that is derived from a seismic acquisition according toembodiments of the invention.

FIG. 9 is an illustration of the energy distribution of atwo-dimensional seismic signal in the frequency-wavenumber domainaccording to an embodiment of the invention.

FIGS. 8, 10, 12, 13 and 14 are flow diagrams depicting multiple scalenoise attenuation techniques according to embodiments of the invention.

FIG. 11 is a plot illustrating a frequency versus wavenumbercharacteristic of a variable length spatial filter according to anembodiment of the invention.

FIG. 15 is a schematic diagram of a data processing system according toan embodiment of the invention.

FIG. 16 is a flow diagram depicting a technique to mitigate edge effectsdue to filtering according to an embodiment of the invention.

DETAILED DESCRIPTION

FIG. 1 depicts an embodiment 10 of a marine-based seismic dataacquisition system in accordance with some embodiments of the invention.In the system 10, a survey vessel 20 tows one or more seismic streamers30 (one exemplary streamer 30 being depicted in FIG. 1) behind thevessel 20. In one non-limiting example, the streamers 30 may be arrangedin a spread in which multiple streamers 30 are towed in approximatelythe same plane at the same depth. As another non-limiting example, thestreamers may be towed at multiple depths, such as in an over/underspread, for example.

The seismic streamers 30 may be several thousand meters long and maycontain various support cables (not shown), as well as wiring and/orcircuitry (not shown) that may be used to support communication alongthe streamers 30. In general, each streamer 30 includes a primary cableinto which is mounted seismic sensors that record seismic signals. Thestreamers 30 contain seismic sensor units 58, which include, inaccordance with embodiments of the invention, multi-component sensors.Each multi-component sensor is capable of detecting a pressure wavefieldand at least one component of a particle motion that is associated withacoustic signals that are proximate to the sensor. Examples of particlemotions include one or more components of a particle displacement, oneor more components (inline (x), crossline (y) and vertical (z)components (see axes 59, for example)) of a particle velocity and one ormore components of a particle acceleration.

Depending on the particular embodiment of the invention, themulti-component seismic sensor may include one or more hydrophones,geophones, particle displacement sensors, particle velocity sensors,accelerometers, pressure gradient sensors, or combinations thereof.

For example, in accordance with some embodiments of the invention, aparticular seismic sensor unit 58 may include at least one particlemotion sensor 70 for purposes of measuring a component of particlemotion along a particular sensitive axis 59 (the x, y or z axis, forexample). As a more specific example, the seismic sensor unit 58 mayinclude a particle velocity sensor that is oriented to acquire ameasurement of a particle velocity along the depth, or z, axis; aparticle velocity sensor to sense a particle velocity along thecrossline, or y, axis; a particle velocity sensor to sense a velocityalong the inline, or x, axis; multiple particle velocity sensors tosense particle velocities along all three (x, y and z) axes; etc.Alternatively, in other embodiments of the invention, the particlemotion sensor(s) of each seismic sensor unit 58 may sense a particlemotion other than velocity (an acceleration, for example).

It is noted that in accordance with some embodiments of the invention,each seismic sensor unit 58 may contain a single component sensor, suchas a pressure sensor. Thus, many variations are contemplated and arewithin the scope of the appended claims.

In addition to the seismic sensor units 58, the marine seismic dataacquisition system 10 also includes one or more seismic sources 40 (twoexemplary seismic sources 40 being depicted in FIG. 1), such as air gunsand the like. In some embodiments of the invention, the seismicsource(s) 40 may be coupled to, or towed by, the survey vessel 20.Alternatively, in other embodiments of the invention, the seismicsource(s) 40 may operate independently of the survey vessel 20, in thatthe source(s) 40 may be coupled to other vessels or buoys, as just a fewexamples.

As the seismic streamers 30 are towed behind the survey vessel 20,acoustic signals 42 (an exemplary acoustic signal 42 being depicted inFIG. 1), often referred to as “shots,” are produced by the seismicsource(s) 40 and are directed down through a water column 44 into strata62 and 68 beneath a water bottom surface 24. The acoustic signals 42 arereflected from the various subterranean geological formations, such asan exemplary formation 65 that is depicted in FIG. 1.

The incident acoustic signals 42 that are created by the source(s) 40produce corresponding reflected acoustic signals, or pressure waves 60,which are sensed by the seismic sensors of the seismic sensor unit 58.It is noted that the pressure waves that are received and sensed by theseismic sensors include “up going” pressure waves that propagate to thesensors without reflection, as well as “down going” pressure waves thatare produced by reflections of the pressure waves 60 from an air-waterboundary, or free surface 31.

The seismic sensors of the seismic sensor units 58 generate signals(digital signals, for example), called “traces,” which indicate theacquired measurements of the pressure wavefield and particle motion. Thetraces are recorded and may be at least partially processed by a signalprocessing unit 23 that is deployed on the survey vessel 20, inaccordance with some embodiments of the invention. For example, aparticular seismic sensor unit 58 may provide a trace, which correspondsto a measure of a pressure wavefield by its hydrophone; and the seismicsensor unit 58 may provide (depending on the particular embodiment ofthe invention) one or more traces that correspond to one or morecomponents of particle motion.

The goal of the seismic acquisition is to build up an image of a surveyarea for purposes of identifying subterranean geological formations,such as the exemplary geological formation 65. Subsequent analysis ofthe representation may reveal probable locations of hydrocarbon depositsin subterranean geological formations. Depending on the particularembodiment of the invention, portions of the analysis of therepresentation may be performed on the seismic survey vessel 20, such asby the signal processing unit 23. In accordance with other embodimentsof the invention, the representation may be processed by a dataprocessing system that may be, for example, located on land or on thevessel 20.

A given seismic measurement contains both desired seismic informationand noise. In the following discussion, the seismic measurement isrepresented by a signal that contains both the desired seismic signalcomponent (i.e., the component indicative of a pressure or particlemotion measurement) and a noise component. The signal may be formed fromraw data that are acquired directly from the seismic sensors, or may beformed by data that are generated by some degree of processing of theraw data, depending on the particular embodiment of the invention.Regardless of whether the signal is acquired directly or not by theseismic sensors, the signal is considered to be derived from a seismicacquisition in the context of this application. Spatial filtering andadaptive noise attenuation-based techniques and systems are disclosedherein for purposes of removing, or attenuating, the noise component tothereby recover the seismic signal component.

As a more specific example, the signal that is derived from a seismicacquisition may contain a relatively low frequency seismic signalcomponent. In this regard, due to the advent of new technology, lowfrequency seismic signals are becoming more important, and modernacquisition geometry allows useful seismic information to be acquired atlower frequencies than what was possible in the past with relativelyshallow towed-streamers. The low frequency content provides deeperpenetration and therefore, improved imaging beneath highly absorptiveoverburdens, such as basalt or salt; less dependency of seismicinversion upon model-based methods; better steep-dip imaging; and bettereffective Q estimation.

Separating the low frequency seismic signal component from the noisecomponent may encounter several challenges, as the noise and seismicsignal components are functions of frequency. In this manner, at lowerfrequencies, the noise is stronger; the seismic signal is weaker; andthe seismic signal has significantly large wavelengths. Conversely, athigher frequencies, the noise is weaker; the seismic signal is stronger;and the seismic signal has shorter wavelengths.

In accordance with embodiments of the invention, the signal may bespatially filtered by a high cut, or low pass, spatial filter toattenuate at least some of the noise. In general, a low pass spatialfilter attenuates relatively high wavenumber (short wavelength) signalsand allows relatively low wavenumber (high wavelength) signals to passthrough, as depicted in an exemplary low pass filter response 100 inFIG. 2. Referring to FIG. 2, in general, the filter has a pass band 102,which allows energy that is associated with a range of relatively smallwavenumbers (i.e., a range of relatively long wavelengths) to becommunicated through the filter at unity or near unity gain. The passband 102 extends from wavenumbers near zero to a cutoff wavenumbercalled “k_(co).” The filter also has a stop band 106, which extends tohigher wavenumbers from a stop band wavenumber, called“k_(sb).” The stopband 106 has an associated stop band attenuation (SBA) 105 andattenuates energy with larger wavenumbers.

Between the pass band 102 and the stop band 106, the filter has atransition band 104, which is defined between the cutoff wavenumberk_(co) and the stop band wavenumber k_(sb). The sharpness of thetransition band 104 is a function of the order, or length, of thefilter. In this regard, a filter with a longer length can be designed tohave a sharper transition (i.e., a more abrupt transition) between thestop band 106 and the pass band 102, as compared to the sharpness of thetransition for a shorter length filter.

As a non-limiting example, in accordance with embodiments of theinvention, the low pass spatial filter may be a finite impulse response(FIR) filter. Some exemplary design techniques that may be employed todesign an FIR filter include window-based methods, Fourier domain-basedsampling techniques, weighted least squares techniques and an equiripplefilter design technique. As a more specific example, in accordance withsome embodiments of the invention, the filter may be designed, ingeneral, pursuant to an equiripple filter technique that is described inJ. H. McClellan, T. W. Parks, and L. R. Rabiner, entitled “A ComputerProgram For Designing Optimum FIR Linear Phase Digital Filters,” IEEETransactions on Audio Electroacoustics, Vol. AU21, pp. 506-526 (1973).It is noted that the optimality criterion in this particular techniqueis the maximum deviation of the desired filter response from the actualfilter response. This is often called the mini-max optimization problemin the literature, because the objective is to minimize the maximumdeviation. The designed filter has equal amplitude deviation from thedesired response in both the pass 102 and stop 106 bands. The relativelevel of the pass 102 and stop 106 band ripples may be controlled byintroducing weights in the design problem. Other filter designtechniques may be used, in accordance with other embodiments of theinvention.

For a towed marine-based seismic acquisition, the apparent velocity ofthe seismic data (in the common shot domain) is constrained to be withina signal cone 180, which is depicted in FIG. 9. In other words, thedesired seismic signal appears inside the signal cone 180. Theboundaries of the signal cone 180 are defined by the acoustic speed ofsound in water. The low pass spatial filter, in accordance withembodiments of the invention, may be designed to attenuate the noiseoutside of the signal cone 180 by selecting the cutoff wavenumber k_(co)as follows:

k _(co) =f/v,   Eq. 1

where “f” represents frequency; and “v” represents the acoustic speed ofsound in water. Referring to FIG. 3, a plot 110 illustrates thevariation of the cutoff wavenumber k_(co) with frequency. In the plot110, the acoustic speed of sound in water was chosen to be 1500 metersper second (m/s), as a non-limiting example.

In accordance with embodiments of the invention, the spatial filteringis conducted so that the order, or length (called “L” herein), of thelow pass spatial filter varies as a function of the frequency beingfiltered, for purposes of accommodating low frequency seismic signals.Because the filter length L varies as a function of frequency, thefilter may be referred to a variable length spatial filter (VLSF). As anon-limiting example, one way to construct a VLSF is to vary thetransition bandwidth (TBW), which is the bandwidth of the transitionband 104 (FIG. 2), proportionally to the signal frequency, as depictedin a plot 116 of the transition bandwidth TBW in FIG. 4. As depicted inFIG. 4, in accordance with some embodiments of the invention, at lowerfrequencies, the “sharpness” of the filter is restricted by defining aminimum transition bandwidth, as indicated at reference numeral 117. Itis noted that other variations are contemplated and are within the scopeof the appended claims. For example, in other embodiments of theinvention, the transition bandwidth may be fixed, and the stop bandattenuation 105 (which is depicted as being a fixed −60 dB attenuationin the non-limiting example shown in FIG. 2) may be varied.

The edge of the stop band, i.e., the stop band wavenumber k_(sb), may bedefined as the sum of the cutoff wavenumber k_(co) and the transitionbandwidth TBW, as follows:

k _(sb) =k _(co) +TBW.   Eq. 2

The length L of the filter, which is required to achieve the desiredstop band attenuation may be described as follows:

$\begin{matrix}{{L = {\frac{( {{\ln \; 2} - {0.05 \times \ln \; 10 \times S\; B\; A}} )}{\pi}\frac{K_{s}}{T\; B\; W}}},} & {{Eq}.\mspace{14mu} 3}\end{matrix}$

where “SBA” represents the stop band attenuation in decibels. Given thevariation in the transition bandwidth TWB with frequency, as illustratedby the plot 116 of FIG. 4, the L filter length also varies withfrequency, as illustrated by a plot 120 in FIG. 5.

Due to the variable filter length, the filtering of a signal derivedfrom a seismic acquisition may be regulated such that the filter lengthchanges with the frequency being filtered. Thus, referring to FIG. 6, atechnique 150 in accordance with embodiments of the invention includesspatially filtering (block 154) a signal that is derived from a seismicacquisition, including varying (block 158) a filter length associatedwith the filtering with the frequency of the signal. Thus, for shorterwavelengths, shorter filter lengths are used; and conversely, when thesignal has a longer wavelength, longer filter lengths are employed.Among the advantages of this technique, better noise attenuation may beachieved, and the signal is preserved at lower frequencies due to theincreased filter length. Additionally, the filtering exhibits anincreased robustness to perturbations at higher frequencies due to theshorter filter length. Other and different advantages are contemplated,in accordance with other embodiments of the invention.

The use of the VLSF has several advantages for purposes of handling edgeeffects. More specifically, because the VLSF filter is an FIR filter (inaccordance with some embodiments of the invention), the edge effects donot propagate into offsets further than the length of the filter. Thelonger filters are only used at the lower end of the frequency spectrumwhere the signal has a relatively long wavelength. For example, at 1Hertz (Hz), the seismic signal has wavelengths longer than 1.5kilometers (km). Furthermore, the VLSF is a linear phase filter, andtherefore, there is no need for forward-backward, or zero-phasefiltering.

In accordance with some embodiments of the invention, the edge effectsmay be handled by first extrapolating the seismic data by using anauto-regressive prediction technique 400 that is depicted in FIG. 16.Referring to FIG. 16, pursuant to the technique 400, an auto-regressivemodel is used (block 404) to extrapolate seismic traces beyond theacquisition geometry in the frequency-space domain. More specifically,the auto-regressive model may be first fit and estimated at a particularfrequency by using, for example, the autocorrelation technique that isdescribed in Steven M. Kay, “Modern Spectral Estimation: Theory andApplication,” (Prentice-Hall 1988). Next, the data is extrapolatedbeyond the acquisition geometry by using the estimated autoregressivemodel. The amount of extrapolation is based on the filter length at thatparticular frequency. The technique 400 subsequently includes performingvariable length spatial filtering on the extrapolated traces, pursuantto block 408. After application of the VLSF, the extrapolated portionsof the traces are discarded. An advantage of this technique is that theautoregressive model may be forced to have a damped response, meaningthat the amplitude of the predicted traces decay at larger offsets. Itis noted that signals with longer wavelengths may be successfullyextrapolated at larger distances. In other words, longer filters areused at lower frequencies, where the signal has longer wavelengths.

Other techniques may be used for purposes of handling edge effects, inaccordance with other embodiments of the invention. For example, inother embodiments of the invention, forward-backward filtering may beused to mitigate edge effects. In this technique, the same filter isapplied twice: once in forward direction in space and then in reversedirection in space. It is noted that the forward-backward filteringeffectively doubles the original length of the filter and may result intransients with longer duration in space.

The transients may be reduced by choosing the initial conditions of theforward and backward filters, as described in Fredrik Gustafsson,“Determining The Initial States In Forward-Backward Filtering,” IEEETransactions on Signal Processing, pp. 988-992, April 1996, Volume 44,Issue 4 and in A. Ozbek and P. Forman, FX-IIR Filter for Seismic Data,U.K. Patent Application 805341.5, which was filed on Mar. 25, 2008. Afurther improvement is possible by the symmetric extension of the databefore the first trace and after the last trace, respectively. Thistechnique is particularly suitable when used with infinite impulseresponse (IIR) filters, due to the fact that IIR filters in general havenon-linear phases; and forward-backward filtering is required to have azero-phase filter response.

As a more specific example, the VLSF may be used in a signal processingtechnique 170 that is depicted in FIG. 7, in accordance with someembodiments of the invention. Pursuant to the technique 170, atime-space domain representation of a signal that is derived from aseismic acquisition is first transformed (block 172) into afrequency-space representation. For each frequency, a filter length isdetermined, pursuant to block 176, which results in the determination ofa corresponding filter that has this determined filter length. Thedetermined filters may then be used to spatially filter thefrequency-space representation of the signal at the associatedfrequencies, pursuant to block 178.

In accordance with some embodiments of the invention, the signal may betransformed into the frequency-space domain using a forward Fouriertransformation, which is described below:

$\begin{matrix}{{{S( {f,x} )} = {\sum\limits_{t}\; {{s( {t,x} )}^{{j2\pi}\; {ft}}}}},} & {{Eq}.\mspace{14mu} 4}\end{matrix}$

where “t” represents time; “x” represents space; “f” representsfrequency; “s(t,x)” represents the time-space domain representation ofthe signal; and “S(f, x)” represents the frequency-space domainrepresentation of the signal. The filter designed pursuant to thetechnique 170 may be represented by “H(f,x),” and the filtering may bedescribed as follows:

G(f,x)=H(f,x)*_(x) S(f,x),   Eq. 5

where “G(f,x)” represents the frequency-space representation of thefiltered data, and “*_(x)” represents spatial convolution. Thetime-space domain representation of the filtered data, called “g(t,x),”may be determined using an inverse Fourier transformation, as describedbelow:

$\begin{matrix}\begin{matrix}{{g( {t,x} )} = {\int{{G( {f,x} )}^{{j2\pi}\; {ft}}{f}}}} \\{{= {h( {t,x} )*{s( {t,x} )}}},}\end{matrix} & {{Eq}.\mspace{14mu} 6}\end{matrix}$

where “*” represents the two-dimensional (2-D) convolution operator; and“h(t, x)” represents the inverse Fourier transformation of thefrequency-space representation of the VLSF filters. In other words, theinverse Fourier transformation h(t, x) is the impulse response of thecorresponding 2-D FIR filter. This means that the VLSF technique may beused to design multi-dimensional FIR filters. The resulting filters maybe applied either in time-space; frequency-space; orfrequency-wavenumber domains, depending on the particular embodiment ofthe invention.

Thus, referring to FIG. 17, in accordance with some embodiments of theinvention, a technique 420 includes providing an n-dimensional FIRfilter that is formed from a plurality of n−1 dimensional FIR filters,pursuant to block 424. A filter that is derived from a seismicacquisition may be spatially filtered in n dimensions using theplurality of n−1 dimensional FIR filters, pursuant to block 428. Forexample, a time-space filter may be designed and applied using spacefilters at every frequency.

The spatial wavenumber content of the signal becomes smaller at lowerfrequencies, and the variable filter length technique described aboveprovides a very sharp filtering response at lower frequencies. However,this filtering technique does not process the signal within theboundaries of the signal cone 180 (FIG. 9). In order to remove noisewithin the signal cone 180, an adaptive noise attenuation technique maybe used in conjunction with the above-described VLSF technique. Althoughit may be conceived that similar to the above-described VLSF-basedtechnique, at each frequency, an adaptive noise attenuation algorithmmay be used for noise attenuation, such an approach would create ringingnoise and artifacts in the time-space domain when used with adaptivefilters. The explanation for this behavior is that for theabove-described VLSF-based techniques, the wavenumber response of thefilters varies smoothly as a function of frequency. However, this smoothvariation is not guaranteed when the filter coefficients at a frequencyare computed as a function of the data at that frequency. As a result,rapid variation may occur among filter coefficients at adjacentfrequencies; and these variations may result in high frequency signaldependent noise in the time-space domain.

As described below, a multiple scale noise attenuation technique may beused in accordance with some embodiments of the invention, for purposesof attenuating noise within the signal cone 180. More specifically, inaccordance with some embodiment of the invention, a technique 190 thatis depicted in FIG. 8 may be used. Pursuant to the technique 190, asignal derived from a seismic acquisition is decomposed (block 192) intosubbands using a transformation such as a Discrete Wavelet Transform(DWT). As described further below, the subbands are decimatedrepresentations of the signal for different subset frequency andwavenumber ranges. The technique 190 includes selectively applying(block 194) adaptive noise attenuation to the subbands to producenoise-attenuated subbands. These noise-attenuated subbands may then beused to reconstruct (block 196) a filtered version of the signal.Additionally, as further described below, in accordance with someembodiments of the invention, the VLSF filtering technique may beapplied during the reconstruction process for purposes of removing noiseoutside of the signal cone 180.

In general, the subbands are higher scale, frequency and wavenumbercomponents of the signal and DWTs. An operation called a DiscreteWavelet Transform (DWT) may be used to decompose the signal into thesubbands. A one-dimensional (1-D) DWT provides a time-frequencyrepresentation of a time signal, by decomposing it into subband signalswith low and high frequency content. Each level of the DWT correspondsto filtering a signal with a low pass filter and a high pass filter andthen decimating the filtered signals. The low and high pass filters aredesigned as perfect reconstruction Quadrature Mirror Filters (QMFs).Mathematically, filtering and decimation operations may be described asfollows:

$\begin{matrix}{{{L^{1}( {n,m} )} = {\sum\limits_{n^{\prime}}\; {{h_{0}( {{2\; n} - n^{\prime}} )}{S( {n^{\prime},m} )}}}},{and}} & {{Eq}.\mspace{14mu} 7} \\{{{H^{1}( {n,m} )} = {\sum\limits_{n^{\prime}}\; {{h_{1}( {{2\; n} - n^{\prime}} )}S( {n^{\prime},m} )}}},} & {{Eq}.\mspace{14mu} 8}\end{matrix}$

where “n” and “m” represents the indices of time and space samples,respectively; “S” represents the input signal; “h₀” and “h₁” representlow and high-pass filters, respectively; and “L¹” and “L¹” represent thesubband signals corresponding to the lower half and higher half of theoriginal spectrum, respectively. If the Nyquist frequency for the inputsignal S is F_(N), then the subband signal L¹ corresponds to therelatively lower frequency band (0−F_(N)/2) of the original spectrum,whereas the subband signal H¹ corresponds to the relatively higherfrequency band (F_(N)/2−F_(N)) of the original spectrum.

The decomposition doubles the sampling interval and halves the frequencycontent. In the following discussion, the subband signals are alsoreferred to as the representation of the signal S at a higher scale. Thescale is denoted by the superscript in the description of the subbandsherein. For example, the subband signal is “scale 1” after first levelof decomposition, which produces the subbands L¹ and H¹.

The decomposition may be repeated to further increase the frequencyresolution. For instance, a second step of the DWT could be used todecompose, for instance, the lower subband signal L¹ into higher scalecomponents in another decomposition stage that is described below:

$\begin{matrix}{{{L^{2}( {n,m} )} = {\sum\limits_{n^{\prime}}\; {{h_{0}( {{{2\; n} - n^{\prime}},m} )}{L^{1}( {n^{\prime},m} )}}}},{and}} & {{Eq}.\mspace{14mu} 9} \\{{L^{1}{H^{1}( {n,m} )}} = {\sum\limits_{n^{\prime}}\; {{h_{1}( {{{2\; n} - n^{\prime}},m} )}{{L^{1}( {n^{\prime},m} )}.}}}} & {{Eq}.\mspace{14mu} 10}\end{matrix}$

In this example, L² corresponds to the frequency band (0−F_(N)/4) of theoriginal spectrum, whereas the subband signal L¹H¹ corresponds to thefrequency band (F_(N)/4−F_(N)/2) of the original spectrum.

It is noted that the signal S may be re-synthesized from subband signalsL¹ and H¹ by upsampling and filtering operations, as described below:

$\begin{matrix}{{{S( {n - d} )} = {{\sum\limits_{n^{\prime}}{{g_{0}( {n - n^{\prime}} )}{{\overset{\_}{L}}^{1}( n^{\prime} )}}} + {\sum\limits_{n^{\prime}}{{g_{1}( {n - n^{\prime}} )}{{\overset{\_}{H}}^{1}( n^{\prime} )}}}}},} & {{Eq}.\mspace{14mu} 11}\end{matrix}$

where “d” represents a constant delay; “g₀” and “g₁” represent low andhigh-pass synthesis filters, respectively; and “ L ¹” and “ H ¹”represent upsampled subband signals which are obtained by padding, orinserting zeros, between the samples, as described below:

$\begin{matrix}{{{\overset{\_}{L}}^{1}(n)} = \{ {\begin{matrix}{{L^{1}( {n/2} )},} & {n\mspace{14mu} {is}\mspace{14mu} {even}} \\{0,} & {otherwise}\end{matrix},{and}} } & {{Eq}.\mspace{14mu} 12} \\{{{\overset{\_}{H}}^{1}(n)} = \{ {\begin{matrix}{{H^{1}( {n/2} )},} & {n\mspace{14mu} {is}\mspace{14mu} {even}} \\{0,} & {otherwise}\end{matrix}.} } & {{Eq}.\mspace{14mu} 13}\end{matrix}$

The 2-D DWT may be obtained by first applying the 1-D DWT in time, asdescribed below:

$\begin{matrix}{{{L^{4}( {n,m} )} = {\sum\limits_{m}\; {{h_{0}( {{{2\; n} - n^{\prime}},m} )}{L^{3}( {n^{\prime},m} )}}}},} & {{Eq}.\mspace{14mu} 14} \\{{{L^{3}{H^{1}( {n,m} )}} = {\sum\limits_{m}\; {{h_{1}( {{{2\; n} - n^{\prime}},m} )}{L^{3}( {n^{\prime},m} )}}}},} & {{Eq}.\mspace{14mu} 15}\end{matrix}$

Next, the 1-D DWT is applied in space, as described below:

$\begin{matrix}{{{L^{4}{\_ L}^{1}( {n,m} )} = {\sum\limits_{m^{\prime}}\; {{h_{0}( {n,{{2\; m} - m^{\prime}}} )}{L^{4}( {n,m^{\prime}} )}}}},{and}} & {{Eq}.\mspace{14mu} 16} \\{{{L^{4}{\_ H}^{1}( {n,m} )} = {\sum\limits_{m^{\prime}}\; {{h_{1}( {n,{{2m} - m^{\prime}}} )}{L^{4}( {n,m^{\prime}} )}}}},} & {{Eq}.\mspace{14mu} 17}\end{matrix}$

where the underscore (“_”) represents that the subband has beendecomposed in space.

As specific examples, if the Nyquist frequency and wavenumber for thesignal S are F_(N) and K_(N), respectively, then the subband signalscorrespond to the following frequency and wavenumber bands: L³H¹: thefrequency band (F_(N)/8−F_(N)/16) and the wavenumber band (0−K_(N));L⁴_H¹: the frequency band (0−F_(N)/16) and the wavenumber band(K_(N)/2−K_(N)); and L⁴_L¹: the frequency band (0−F_(N)/16) and thewavenumber band (0−K _(N)/2)

In accordance with embodiments of the invention, the above-describeddecomposition may be used in a multiple scale noise attenuationtechnique 200 that is depicted in FIG. 10. Referring to FIG. 10, thetechnique 200 involves the use of decomposition stages 202 andreconstruction stages 220. For the decomposition stages 202, adaptivenoise attenuation 206 is applied to decomposed subbands at differentstages of the decomposition process. As specific non-limiting examples,the adaptive noise attenuation may be pursuant to a technique similar tothe one disclosed in U.S. Pat. No. 6,446,008, entitled, “AdaptiveSeismic Noise And Interference Attenuation Method,” which issued on Sep.3, 2002; or the technique disclosed in U.S. Pat. No., 6,651,007,entitled, “Adaptive Seismic Noise and Interference Attenuation Method,”which issued on Nov. 18, 2003. The decomposition of the signal into thesubbands may be performed in time (via the exemplary stages 204 a, 204 band 204 c); and in time and space (as depicted by exemplary stages 208 aand 208 b).

The decomposition occurs successively, in a stage-by-stage process. Inthe initial stage 204 a, the original signal is decomposed into the Hand L¹ subbands. The resulting L¹ subband is received by the stage 204 bfor further decomposition, and the H subband is furnished to one of thereconstruction stages 220. The L¹ subband, in turn, is received by thenext decomposition stage 204 b, which decomposes the L¹ subband into theLH subband and the L² subband. The LH subband is furnished to one of thereconstruction stages 220, and the L² subband is furnished to the nextdecomposition stage 204 c. Stage 204 c, in turn, decomposes the L²subband into an L³ subband and the L²H subband, which is furnished toone of the reconstruction stages 220.

As can be seen from FIG. 10, initially, the decomposition stages 202produce more decimated and lower frequency subbands which propagatethrough the decomposition stages 202, ultimately resulting in thegeneration of the L³ subband. The higher frequency subbands, which areproduced in the initial decomposition, in turn, are furnished to thereconstruction stages 220.

Adaptive noise attenuation is not performed, in accordance with someembodiments of the invention, until the L³ subband is generated. This isdue to the fact that the noise is usually weak at higher frequencies,and the adaptive noise attenuation techniques, which are usuallycomputationally intensive, do not have much benefit at these higherfrequencies. Therefore, the adaptive noise attenuation, in accordancewith the example depicted in FIG. 10, begins with the L³ subband andcontinues with the lower frequency subbands that are produced therefrom.

More specifically, a decomposition stage 206 a performs adaptive noiseattenuation on the L³ subband and furnishes the resultantnoise-attenuated subband to the next decomposition stage 208 a. Thedecomposition stage 208 a, in turn, decomposes the L³ subband pursuantto time and space. In other words, the decomposition stage 208 aproduces a decimated lower frequency and lower wavenumber subband L⁴_Lsubband, which is processed by the next adaptive noise attenuation stage206 b. The stage 208 a also produces a higher frequency and higherwavenumber subband L³H, which is furnished to one of the reconstructionstages 220. The L⁴_L subband, in turn, is processed by the adaptivenoise attenuation stage 206 b, and the resulting noise-attenuatedsubband is furnished to the next decomposition stage 208 b.

The above-described process may continue for several successivefrequency and wavenumber decompositions, beyond the exemplary stagesthat are depicted in FIG. 10. Thus, further decompositions in time andspace occur; and each lower frequency and lower wavenumber subband isprocessed by another adaptive noise attenuation stage and then, providedto another decomposition stage 208; and the higher frequency and higherwavenumber subbands produced by the decomposition are furnished to thereconstruction stages 220.

Each splitting operation reduces the data size by two in time and space.Therefore, the number of splits in time and space is limited by therecord length in time than the streamer length in space. If a low numberof splits is performed, the filters may have a “fuzzy” low frequencyresponse. On the other hand, because of the relatively weak signalcontent at very low frequencies, the splitting may be stopped when thefrequency content of the subband signal drops below 4 Hertz.

The reconstruction stages 220 receive the noise-attenuated subbands fromthe decomposition stages 202 and use these subbands to reconstruct asignal in which noise attenuation has been performed in the signal cone180 (FIG. 9). The reconstructions stages 220 include time and spacereconstruction stages 222 (exemplary stages 222 a and 222 b, depicted asexamples in FIG. 10) and time only reconstruction stages 224 (exemplarystages 224 a, 224 b and 224 c, being depicted as examples), whichcorrespond to their respective counterparts in the decomposition stages202. The subbands provided by the decomposition stages 202 are used bythe reconstruction stages 222 and 224 to reconstruct the signal.

Additionally, in accordance with some embodiments of the invention,VLSF-based filtering stages 226 (VLSF-based filtering stages 226 a, 226b, 226 c, 226 d, 226 e and 226 f, being depicted as examples), areinterdispersed among the reconstruction stages 220 for purposes ofattenuating noise outside of the signal cone 180. Each VLSF-based stage226 is a low pass spatial filter, which has a filter length that isbased on the frequencies of the particular subband being filtered. Forexample, the VLSF-based stage 226 d has the filter length that is basedon a frequency range associated with the L² subband, the VLSF-basedstage 226 b has a filtering length based on the frequency rangeassociated with the L⁴_L subband, etc.

It is noted that when 2-D QMF splitting is performed, the scales of thetime and space components are increased by the same factor. With eachadditional split, a higher scale representation of the signal isobtained by zooming into lower frequencies and wavenumbers. Because ofthis property, a single adaptive algorithm prototype may be designed atone scale (a scale that corresponds to the scale of the subband L³, forexample) and used for signals at other scales. For example, if theadaptive noise attenuation technique that is described in U.S. Pat. No.6,446,008 is used, the same number of time and space coefficients andthe same generalized constraints may be used at every scale. Similarly,the VLSF coefficients may be designed at one stage, and the samecoefficients may be used at other higher or lower stages. The largerapertures of the fixed and adaptive filters at higher scales areobtained by a larger sampling interval and spacing rather than by largernumbers of filter coefficients.

In accordance with some embodiments of the invention, the H₀ and H₁ areFIR half band filters, which are not “perfect” low and high passfilters. Therefore, each filtering and decimation operation createsaliasing at higher frequencies. When shorter analysis filters are used,the aliased components may leak outside of the signal cone 180. Forpurposes of ensuring that the noise attenuation techniques do not removean aliased component if the aliased component partially appears outsideof the signal cone, the QMF synthesis filters are designed to put thealiased component back into place when the low and high frequencysubband signals are synthesized into a lower scale signal. For thisreason, in accordance with some embodiments of the inventions, the VLSFcoefficients are designed so that the frequency-wavenumber responses ofthe filters flare out close to the Nyquist wavenumber of the subbandsignal, as depicted in an exemplary plot 230 of the frequency versuswavenumber characteristic of an exemplary VLSF filter in FIG. 11. Asshown, below the Nyquist wavenumber, the filter has a first portion 232that generally follows the slope of the signal cone 180, and beyond theNyquist wavenumber, the filter has a smaller slope 234 to ensure thecapture of aliased components.

The QMF splitting and reconstruction operations may be performed bycircular convolution. This usually creates artifacts on finite lengthdata. A technique used to mitigate edge effects may include process ofdata to make the data circularly symmetric in time and space before theQMF splitting. For purposes of making a 2-D dataset circularlysymmetric, the dataset may be extended in time and space by symmetricreflection about the last trace and time sample. It is noted that inaccordance with other embodiments of the invention, the above-disclosedtechnique of using extrapolation may be used for purposes of addressingthe edge effects. In some other applications, the data may be made morecircularly symmetric in time and extrapolated in space.

It is noted that with a properly selected shot window, the seismicsignal may be located away from the start and end of the shot window. Inthat case, the circular convolution effects appear only in the noisepart of the time-space plots; and therefore, the symmetric extension intime may be omitted. However, extrapolation in space may still be usedto avoid edge effects due to both QMF splitting in space and theapplication of the VLSF filters afterwards.

Other embodiments are contemplated and are within the scope of theappended claims. For example, referring to FIG. 12, in accordance withother embodiments of the invention, an alternative multiple scale noiseattenuation technique 240 may be used. In general, the technique 240 issimilar to the technique 200 (FIG. 10), except that the reconstructionstages 242 (which replace the reconstruction stages 220) do not employVLSF filtering.

As another variation, a technique 250, which is depicted in FIG. 13, maybe used in accordance with other embodiments of the invention. In thetechnique 250, adaptive noise attenuation is not performed. Morespecifically, pursuant to the technique 250, a signal that is derivedfrom a seismic acquisition is decomposed into subbands, pursuant toblock 252. The signal is both reconstructed (block 254) from thesedecomposed subbands; and the decomposed subbands are spatially filtered(pursuant to the herein disclosed VLSF filtering technique, forexample), pursuant to block 256. Thus, for each subband, a length of thefiltering is regulated based on a frequency associated with the subband,pursuant to block 258.

As a more specific example, a multiple scale noise attenuation technique270 that is depicted in FIG. 14 may be used in accordance with someembodiments of the invention. In general, the technique 270 is similarto the technique 200 (FIG. 10), in that the reconstruction stages 220are the same. However, unlike the technique 200, the technique 270includes decomposition stages 274, which replace the decompositionstages 202 and do not include the adaptive noise attenuation stages 206.

Other variations are contemplated and are within the scope of theappended claims.

Referring to FIG. 15, in accordance with some embodiments of theinvention, a processing system 320 may perform at least part of one ormore of the techniques that are disclosed herein, such as techniquesrelated to variable length spatial filtering, FIR filter design, edgeeffect mitigation, adaptive noise attenuation and multiple scale noiseattenuation, as non-limiting examples.

The system 320 may be located on one of the streamers 30, on eachstreamer 30, distributed among the streamers 30, on the seismic source40, on the survey vessel 20, at a remote land-based facility, etc. Thesystem 320 may also be distributed on one or more of these entities, inaccordance with other embodiments of the invention. In accordance withsome embodiments of the invention, the system 320 may include aprocessor 350, such as one or more microprocessors and/ormicrocontrollers.

The processor 350 may be coupled to a communication interface 360 forpurposes of receiving seismic data, which are indicative of seismicmeasurements. In other words, the seismic data indicates signals thatare derived from a seismic acquisition and have associated noisecomponents as well as components that are indicative of pressuremeasurements, vertical particle motion measurements and crosslineparticle motion measurements, etc. In this regard, the seismic data maybe indicative of the raw data received from the streamers 30, processedseismic data, etc. Regardless of its particular form, the seismic datais indicative of at least one signal that is derived from a seismicacquisition.

As a non-limiting example, the interface 360 may be a USB serial businterface, a network interface, a removable media (such as a flash card,CD-ROM, etc.) interface or a magnetic storage interface (IDE or SCSIinterfaces, as examples). Thus, the interface 360 may take on numerousforms, depending on the particular embodiment of the invention.

In accordance with some embodiments of the invention, the interface 360may be coupled to a memory 340 of the system 320 and may store, forexample, various input and/or output data sets 348 involved with thetechniques that are described herein. The memory 340 may store programinstructions 344, which when executed by the processor 350, may causethe processor 350 to perform at least part and possibly all of one ormore of the techniques that are described herein and display resultsobtained via the technique(s) on the display 374 of the system 320, inaccordance with some embodiments of the invention. As shown in FIG. 15,the system 320 may include a display interface 370 that couples thedisplay device 374 to the system 320.

Other embodiments are contemplated and are within the scope of theappended claims. For example, although a towed marine-based seismicacquisition system has been disclosed, the techniques and systems thatare disclosed herein may likewise be applied to signals that are derivedfrom other types of seismic acquisitions, such as land-based seismicacquisitions, borehole-based seismic acquisitions, seabed-based seismicacquisitions, etc.

While the present invention has been described with respect to a limitednumber of embodiments, those skilled in the art, having the benefit ofthis disclosure, will appreciate numerous modifications and variationstherefrom. It is intended that the appended claims cover all suchmodifications and variations as fall within the true spirit and scope ofthis present invention.

1. A method comprising: spatially filtering a signal derived from aseismic acquisition, the filtering being associated with a filterlength; and varying the filter length with frequency.
 2. The method ofclaim 1, further comprising varying a transition bandwidth associatedwith the filtering based on the frequency being filtered.
 3. The methodof claim 1, further comprising varying a stop band attenuationassociated with the filtering based on the frequency being filtered. 4.The method of claim 1, wherein the acts of spatially filtering thesignal and varying the filter length comprise: providing a time-spacedomain representation of the signal; transforming the time-space domainrepresentation into a frequency-space representation; for eachfrequency, determining a filter length and determining a filtercorresponding to the determined filter length; and for each frequencyusing the determined filter to spatially filter the frequency-spacerepresentation of the signal.
 5. The method of claim 3, furthercomprising: transforming the spatially filtered frequency spacerepresentation of the signal into a time domain representation of aspatially filtered signal.
 6. The method of claim 1, wherein the acts ofspatially filtering the signal and varying the filter length comprise:decomposing the signal in time to produce subbands, each subband beingassociated with at least different frequency ranges of the signal; foreach frequency range, determining a filter length and determining afilter corresponding to the determined filter length; and for eachfrequency range, using the determined filter to spatially filter thesignal.
 7. The method of claim 1, wherein the signal is associated witha given acquisition geometry and the acts of spatially filtering thesignal and varying the filter length comprise: providing a first signalindicative of measurements acquired in a seismic survey and beingassociated with a given acquisition geometry; extrapolating the firstsignal to generate a second signal indicative of first measurementsconfined by the geometry and second measurements not confined by thegeometry; spatially filtering the second signal; and after the spatialfiltering of the second signal discarding a portion of the filteredsecond signal associated with the second measurements.
 8. A methodcomprising: decomposing a signal derived from a seismic acquisition intosubbands in successive stages, the subbands being associated with atleast different frequency ranges of the signal; selectively applyingadaptive noise attenuation in between the successive stages such thatthe stages decompose noise-attenuated subbands; and reconstructing thesignal from the subbands resulting from the decomposition.
 9. The methodof claim 8, wherein some of the subbands resulting from thedecomposition are associated with lower frequencies than the remainingsubbands resulting from the decomposition; and the act of applyingadaptive noise attenuation comprises applying the adaptive noiseattenuation to the subbands that are associated with the lowerfrequencies and not applying adaptive noise attenuation to the remainingsubbands.
 10. The method of claim 8, wherein the subbands are furtherassociated with different wavenumber ranges.
 11. The method of claim 10,wherein some of the subbands resulting from the decomposition areassociated with lower wavenumbers than the remaining subbands resultingfrom the decomposition; and the act of applying adaptive noiseattenuation comprises applying the adaptive noise attenuation to thesubbands that are associated with the lower wavenumbers and not applyingthe adaptive noise attenuation to the remaining subbands.
 12. The methodof claim 8, wherein some of the subbands resulting from thedecomposition are within a frequency-wavenumber region associated with adesired seismic signal; and the act of applying adaptive noiseattenuation comprises applying the adaptive noise attenuation to thesubbands that are within the frequency-wavenumber region and notapplying the adaptive noise attenuation to the first subbands that areoutside of the frequency-wavenumber region.
 13. The method of claim 8,wherein the act of decomposing comprises performing discrete wavelettransformations.
 14. The method of claim 8, further comprising:spatially filtering the noise attenuated second subbands; and for eachsecond subband, regulating a length of the filtering based on afrequency associated with the second subband.
 15. The method of claim14, further comprising: adapting the filtering to pass through aliasingcomponents of the noise-attenuated subbands.
 16. A method comprising:decomposing a signal derived from a seismic acquisition into subbands,the subbands being associated with at least different frequency ranges;reconstructing the signal from the subbands; spatially filtering thesubbands; and for each subband, regulating a length of the filteringbased on the frequency subband associated with the subband.
 17. Themethod of claim 16, wherein the act of decomposing is performed insuccessive stages, each stage decomposing an input subband to produce afirst output subband used for the reconstruction and a second outputsubband used as the input subband for the next successive stage.
 18. Themethod of claim 16, wherein the act of reconstructing is performed insuccessive stages, each stage reconstructing an output subband based ona subband provided by the decomposing and an input subband provided asthe output subband by another one of the stages.
 19. A systemcomprising: an interface to receive data indicative of a signal derivedfrom a seismic acquisition; and a processor to: process the data tospatially filter the signal, the filtering being associated with afilter length; and vary the filter length with frequency.
 20. The systemof claim 19, wherein the processor is adapted to process the data to:provide a time-space domain representation of the signal; transform thetime-space domain representation into a frequency-space representation;for each frequency, determine a filter length and determine a filtercorresponding to the determined filter length; and for each frequencyuse the determined filter to spatially filter the frequency-spacerepresentation of the signal.
 21. The system of claim 19, wherein theprocessor is adapted to process the data to: decompose the signal intime to produce subbands, each subband being associated with at least adifferent frequency range of the signal; for each frequency range,determine a filter length and determine a filter corresponding to thedetermined filter length; and for each frequency range, use thedetermined filter to spatially filter the signal.
 22. A systemcomprising: an interface to receive data indicative of a signal derivedfrom a seismic acquisition; and a processor to: decompose the signalinto subbands in successive stages, the subbands being associated withat least different frequency ranges of the signal; selectively applyadaptive noise attenuation in between the successive stages such thatthe stages decompose noise-attenuated subbands; and reconstruct thesignal from the subbands resulting from the decomposition.
 23. Thesystem of claim 22, wherein the processor is further adapted toselectively spatially filter the subbands between the stages andregulate a length of the filtering based on a frequency associated withthe subband being filtered.
 24. The system of claims 22, wherein some ofthe subbands resulting from the decomposition are associated with lowerwavenumbers than the remaining subbands resulting from thedecomposition, and the processor is further adapted to process the datato apply the adaptive noise attenuation to the subbands that areassociated with the lower wavenumbers and not apply the adaptive noiseattenuation to the remaining subbands.
 25. The system of claim 22,wherein some of the subbands resulting from the decomposition are withina frequency-wavenumber region associated with a desired seismic signal;and the processor is further adapted to apply the adaptive noiseattenuation to the subbands that are within the frequency-wavenumberregion and not apply the adaptive noise attenuation to the subbands thatare outside of the frequency-space region.
 26. The system of claim 22,wherein the processor is further adapted to perform discrete wavelettransformations to decompose the signal.
 27. The system of claim 22,further comprising: a spread of at least one streamer comprising seismicsensors to acquire data indicative of the signal derived from theseismic acquisition; and a vessel to tow the spread.
 28. A systemcomprising: an interface to receive data indicative of a signal derivedfrom a seismic acquisition; and a processor to process the data to:decompose the signal into subbands, the subbands being associated withat least different frequency ranges; reconstruct the signal from thesubbands; spatially filter the subbands; and for each subband, regulatea length of the filtering based on the frequency range associated withthe subband.
 29. The system of claim 28, wherein the processor isfurther adapted to decompose the signal in successive stages, each stagedecomposing an input subband to produce a first output subband used forthe reconstruction and a second output subband used as the input subbandfor the next successive stage.
 30. The system of claim 28, wherein theprocessor is further adapted to process the data to perform thereconstruction in successive stages, each stage reconstructing an outputsubband based on a subband provided by the decomposition and an inputsubband provided as the output subband by another one of the stages. 31.The system of claim 28, further comprising: a spread of at least onestreamer comprising seismic sensors to acquire data indicative of thesignal derived from the seismic acquisition; and a vessel to tow thespread.
 32. A method comprising: filtering a signal derived from aseismic acquisition; and mitigating edge effects in connection with thefiltering, the mitigating comprising using autoregressive modeling toextrapolate the signal beyond an acquisition geometry to form anextrapolated signal and performing variable length spatial filtering onthe extrapolated signal.
 33. The method of claim 32, wherein the act ofusing autoregressive modeling comprises fitting and estimating anautoregressive model at a given frequency.
 34. A method comprising:providing an n dimensional FIR filter formed from a plurality of n−1dimension FIR filters; and spatially filtering a signal derived from aseismic acquisition in n dimensions using the plurality of n−1dimensional FIR filters to generate a filtered signal.
 35. The method ofclaim 34, wherein the signal that is spatially filtered indicatesgeological formation properties.